Lambert will undergo a medical on Saturday before putting the finishing touches to a two-year deal and flying out to Miami with England's World Cup squad.

The Anfield club's previous bid for Lallana, worth around £20m, was turned down but the player is thought to be keen to join Liverpool, who will be in the Champions League next season. They hope Friday night's offer will be enough to persuade Southampton to sell. Lambert, 32, is understood to have become increasingly disillusioned with life at Southampton, despite another impressive season, and is keen to pursue a new challenge.

Lambert spent five years as a youth player with Liverpool, and will return to the club he supported as a boy with the prospect of Champions League football next season.

Southampton agreed a fee of around £4m yesterday, with the total likely to increase on appearances. It has emerged that Liverpool warned them that the deal would be scrapped if it had not gone through before the flight to Miami.

Southampton had been determined to hold out for a figure close to £30m for Lallana, but must now decide whether £25m is enough. Liverpool will tell Southampton the latest bid represents their final offer.

The fee would represent a significant chunk of Rodgers's transfer budget, and the deal has been further complicated by interest from rival clubs, believed to include Manchester United and Arsenal.

Roy Hodgson, the England manager, has warned that he will not tolerate any of his players being distracted by potential transfers in the build-up to the World Cup in Brazil.

Hodgson will allow medical staff from clubs to fly out to Miami, however, which leaves Liverpool facing a crucial week to try and thrash out a deal for Lallana.

The exit of Lambert is raising fears of a possible player exodus at St Mary's, with Manchester United still locked in negotiations over a £30m deal for defender Luke Shaw. United are confident of signing the left-back next week but, like Liverpool, are struggling to agree on the final fee, despite extensive talks.

Dejan Lovren, the Southampton centre-back, is another target for Liverpool, while there are also doubts over the futures of Jay Rodriguez and Morgan Schneiderlin. Manager Mauricio Pochettino left Southampton for Tottenham Hotspur earlier this week and is expected to target a number of his former players. Spurs have made an inquiry for Lallana, but there are doubts over whether chairman Daniel Levy will agree to pay the required £30m.

Southampton, meanwhile, will step up their search for Pochettino's successor next week, with former Holland international Ronald Koeman registering his interest. Koeman will leave Feyenoord this summer, and his representative, Guido Albers, said: "From Ronald's point of view it would be amazing. I think it would be a very good success for the club, but also for the Premier League to get a coach like him. He would like to work there."